Pallas I don't have an acoustic piano. But when I researched recording for acoustic, I came to same conclusion as @Gombessa the small-diaphragm condenser microphone is best suited to record piano, and the Behringer C-2 set is a really good option.
Since this is for a semi-permanent in-home setup, I would recommend skip the portable recorders, and record directly into your computer. You will have a larger screen, better interface to adjust settings, live preview, and ability to instantly review the recordings.
The equipment you need to connect your microphone to your computer is a USB Audio Interface. The most popular one is Focusrite Scarlett 2i2. A less expensive alternative, which I have, is Behringer UMC202HD. You don't need a mixer, the mixing can be done via software on your computer. But if for any other reason you are looking for a mixer that can also work as USB audio interface, then I strongly recommend Yamaha AG06.
As for software, my first recommendation is to use OBS Studio. It can record multiple audio and video streams. You have a lot of flexibility with video composite layout, and even the ability to add overlays (images, live clock, etc). Multiple audio tracks can be mixed into single track, and/or saved to individual audio tracks. Although most video players on PC/Mac can only playback audio track 1, you can extract the additional audio tracks using editing software.
For audio editing, Audacity seems to be everyone's go to tool. It's free and it's really good. Or you can use Adobe Audition. Adobe Premiere is a good video editing tool. l learned it first. However, there is an alternative, Davinci Resolve, that worth considering, too. Davinci Resolve is like Adobe Premiere, Adobe After Effect, and Adobe Audition rolled into one software, with a more user friendly interface. Since I started using Davinci Resolve, I have not touched Adobe Premiere for a long time.
Oh, cameras. Try some good webcams first. They have come a long way from the old days. They even come in 4K resolution nowadays, which gives you freedom to crop the footage later in editing. If your lighting is good, webcam can produce perfectly adequate image quality. I have some Logitech 920/930, and an Elgato Facecam. And I have not felt any need to switch them to my digital cameras (which I have several). The hassle with cameras is you need to figure out how to power them for a permanent setup, while not killing their batteries (by charging them all the time). Also larger cameras need stronger support, which makes them harder to place for higher angle shots. Web cams are so light, you can put them on microphone stands, or articulating arms (using mini ball head) - all much cheaper than camera tripods. At least consider webcams for your 2nd or 3rd angle shots. For your main angle, if you choose to use a DSLR or mirrorless, look up your camera manufacture's website and see if they have released a software that allows connecting the camera as a webcam (live camera) to your computer. This way, your camera can feed to video to OBS Studio, too.